Biomedical Engineering Reference
In-Depth Information
where
M
1
and
M
2
are the rest masses and
z
1
and
z
2
are the charges. If particle
number 2 is a proton (
M
2
= 1
and
z
2
= 1
), then we can write for the range
R
of the
other particle (mass
M
1
=
M
proton masses and charge
z
1
=
z
)
M
z
2
R
p
(
R
(
β
)
=
β
),
(5.43)
where
R
p
(
β
)
is the proton range.
Example
Use Table 5.3 to find the range of an 80-MeV
3
He
2+
ioninsofttissue.
Solution
Applying (5.43), we have
z
2
= 4,
M
= 3, and
R
(
β
) = 3
R
p
(
β
)/4. Thus the desired range
is three-quarters that of a proton traveling with the speed of an 80-MeV
3
He
2+
ion. At
this speed, the proton has an energy of 80/3
26.7 MeV, that is, an energy smaller
than that of the helium ion by the ratio of the masses. Interpolation in Table 5.3 gives
for the proton range at this energy
R
p
=
=
0.705 g cm
-2
. It follows that the range of the
80-MeV
3
He
2+
particle is (
4
)(0.705)
0.529 g cm
-2
, or 0.529 cm in unit-density soft
=
tissue.
Figure 5.7 shows the ranges in g cm
-2
of protons, alpha particles, and electrons
in water or muscle (virtually the same), bone, and lead. For a given proton en-
ergy, the range in g cm
-2
is greater in Pb than in H
2
O, consistent with the smaller
mass stopping power of Pb, as mentioned at the end of Section 5.6. The same
comparison is true for electrons in Pb and water at the lower energies in Fig. 5.7
(
20 MeV). At higher energies, bremsstrahlung greatly increases the rate of energy
loss for electrons in Pb, reducing the range in g cm
-2
below that in H
2
O.
Figure 5.8 gives the range in cm of protons, alpha particles, and electrons in air
at standard temperature and pressure. For alpha particles in air at 15
◦
C and 1-atm
pressure, the following approximate empirical relations
3)
fit the observed range
R
in cm as a function of energy
E
in MeV:
R
=
0.56
E
,
E
< 4;
(5.44)
R
=
1.24
E
- 2.62,
4 <
E
< 8.
(5.45)
Alpha rays from sources external to the body present little danger because their
range is less than the minimum thickness of the outermost, dead layer of cells of
the skin (epidermis, minimum thickness
∼
7mgcm
-2
). The next example illus-
trates the nature of the potential hazard from one important alpha emitter when
inhaled and trapped in the lung.
3 U.S. Public Health Service,
Radiological
Health Handbook
, Publ. No. 2016, Bureau of
Radiological Health, Rockville, MD (1970).
